Representing a new chapter in condensed matter physics and material science, topological quantum states [1]}, [2]} have attracted the wide attention of researchers since their discovery. The topological concepts of electronic structures have been widely predicted and verified  [3]}, [4]}, [5]}, [6]}, [7]}, [8]}, [9]}, [10]}, [11]}, [12]}, [13]}, [14]}. To date, thousands of topological electronic materials have been widely predicted and experimentally observed  [15]}, [16]}, [17]}, [18]}, [19]}, [20]}, [21]}, [22]}, [23]}, [24]}, [25]}. Moreover, phonons, the most basic emergent boson of crystalline lattices, are the energy quantum of lattice vibration. Phonons make important contributions to the thermal conductivity, and specific heat of non-metal. The coupling between phonons and electrons determines the properties of many physical phenomena, including superconductivity, thermoelectric effect and carrier mobility [26]}, [27]}. By analogy with well-studied electronic systems, topological concepts have been introduced to the field of phonons and named topological phonons [28]}, [29]}, [30]}, [31]}, [32]}, [33]}. It is worth mentioning that unlike electrons, phonons are bosons and are not restricted by the Pauli exclusion principle, so we can focus on the topological signatures of a wide frequency range.
